Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.